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Anyone who has studied Feynman's path integral will know that it makes quantum mechanics more like classical mechanics. A student who has learned about the Lagrangian will easily understand the concept of quantum mechanics through the path integral (I think). Why do we still keep using Schrödinger's approach? It's history now.
Maybe to fully understand the Feynman's approach, you need to, erm, integrate so much? I mean, maybe it is based heavily on math?
I do not agree with the statement, that the lack of mathematical rigor is a major reason for not teaching the path integral formalism in quantum mechanics. The common physicist is normally not interested in complete mathematical rigor, as long as the concepts make sense from a physical point of view and produce the right results. A good example of this is the Dirac formalism - every physicist knows it, and everyone with an elementary education in functional analysis knows that the mathematical justification cannot rely on pure Hilbert space theory. I don't know any textbook which even does an attempt to formulate Dirac's formalism in a rigorous way (Apart from that usual gibberish about spectral decompositions in finite dimensions, which does of course not apply to most problems in QM), that is because the physicist doesn't care and the mathematical physicist does not use bras and kets. The mathematical formulation needs, apart from spectral theory, also some stuff about locally convex and nuclear spaces, which is also pretty specific.
The Feynman integral is a quite intuitive construct from a purely physical point of view, and everyone taking a course in quantum mechanics should have basic education in analytical mechanics, at least it's like this in Germany - usually Lagrange and Hamilton is covered, sometimes also a bit of Hamilton-Jacobi. I don't think this is too much to be asked from an undergrad student who had a course on mechanics. Furthermore, I also don't think that it is too hard for an undergrad to use it, therefore the skills required to introduce and work with the path integral should be given.
From my point of view, there are three major reasons, for the Schrödinger equation being the "standard approach" to quantum mechanics, and not the path integral formulation. Of course, these are correlated.
The first reason is of course, that it has always been done like this. For this reason, there exist a lot of good and established textbooks using this approach, whom I can base a lecture on. There is a wide range of good material and problems. Furthermore, depending on in which area you will work in, the Schrödinger approach is usually a good approach to work with, therefore it is by far not outdated - path integrals are primarily used by particle physicists, in other areas, Schrödinger is usually very useful.
The second reason is, that simple problems one can imagine are usually easier to solve using the Schrödinger equation rather than the path integral formulation. For example, try to compute the harmonic oscillator using path integrals. It's complicated, not elegant and you do not get a big physical insight in what happens. The hydrogen atom is even worse, and I don't want to start with Delta potentials. Another major disadvantage, which makes it quite complicated to solve certain problems, is that it is difficult to change representation in order to compute something. The Feynman formalism deals well with certain problems which primarily appear in QFT, but no one in his right mind would try to solve the hydrogen atom using path integrals. Of course, the choice of what we call "simple and standard problems" is affected by the way we teach quantum mechanics, so this point is correlated to the first one.
The third reason is, that the Schrödinger approrach (in my opinion) has more educational value. Firstly, because it is an approach we in principle already know from mechanics and electrodynamics: Here you have a differential equation, go solve it. In mechanics, it's Newton/Lagrange/Hamilton, in electrodynamics its Maxwell, and in quantum mechanics its the Schrödinger equation. Furthermore, while solving the SE, you usually solve the eigenvalue problem for the Hamiltonian, which of course has a very direct and experimentally observable interpretation as the energy states of the respective system. The kernel of time evolution is a rather abstract construct, which does not have a direct interpretation like this.
I would like to add a slightly more technical answer to supplement the other (although it is a bit late). It may be helpful, I think, and it touches a broader issue which interesting. It's a conflict of history of teaching versus history of the mathematical prerequisites for the material.
First of all, supposing the topic material is taught early on, how would that work? I like the following books:---they benefit from the long experience of the authors in teaching this material.
Cartier P, DeWitt-Morette C, 2006, Functional Integration, Cambridge University Press. (B)
DeWitt B, 2003, Global Approach to Quantum Field Theory, I, II, Oxford University Press. (C)
Prior this, the following textbook would introduce all basics, for example to student starting with basic knowledge of calculus.
Choquet-Bruhat Y, DeWitt-Morette C, 1982, Analysis, Manifolds, and Physics, I, Elsevier. (A)
This would be a selfcontained series of courses: A, B, C, which would also teach a good chunk of other material, so a pair of introductory mathematical courses, whose content is covered in (A) would be dropped. So, only one more course, in total, would be required. Why isn't this happening?
(X) The answer is indeed historical, due to historically the prerequisite mathematics for the subject, when first developed, gaining the reputation of being "very advanced material" with few introductory texts, and this reputation stuck. Even when no longer justified, it influences how departments construct their curriculum. Mathematics is synergistic. What was once difficult becomes easier when other concepts are discovered and developed. Advances in method over time made the calculations easier and no longer really something difficult in the usual instances, and plenty of introductory books to the material now exist. However, reputations change more slowly than the environment that generated them.
For example, A.E. above brought up the point that analytic continuation is not taught to undergraduates in many places, and complex variables aren't mandatory. That's my experience too. I suggest the same circumstances being the reason.
For rigorous work with functional integration, distribution methods must be used. (Now I prefer sheaf cohomology methods leading to generalized functions. But the justification and application is the same in this case.) The historical issue is: category theory, sheaves, and generalized functions became widely known and fully developed in the 1970's. (After Feynman's integral was in use for over twenty years!) To know how to do calculations in the arbitrary case, beyond the intuitive setting up of the problems (which was developed 1920's-1950's), these later methods are required.
There are plenty of basic introductory books, published 1990's-present, which are not too advanced for undergraduates. E.g., Lawvere's book, MacLane's several books, etc., assume the reader begins with no knowledge whatsoever about even the most elementary mathematical concepts! And the introductory books on generalized functions and distributions begin easily: right hand limits of appropriate series of sums and products applied to 19th century problems, to solve them more easily than is traditional.
The issue is that they are, in fact, books. Buying a book is not sufficient to learn anything. It must be read. There often (for various institutional reasons) simply isn't enough time in the major curriculum to present all the material without having the undergraduate read something in some book that is not presented in class. But in most universities, at this level, use books, if at all, mostly for homework problems... :_( Historically, the education system has been drifting away from required reading the STEM fields.
On the other hand, the other methods can be presented without required reading. And while the typical undergraduate, coming from a typical high school can read books, they usually don't. If they buy them, it's because they contain required homework problems. Many undergraduate courses are not structured to require reading at all. The textbooks are selected only for their changing homework sets.
As you can imagine, and as is indeed the case, whatever cannot be taught this way, or has the reputation of "advanced" isn't mandatory or taught at all until well into graduate coursework. This is merely one case, it seems to me.
Let me try to summarize what was said in the comments and add something. (I am a teacher of mathematics, not physics, but this is close). First of all not all students study Lagrangian mechanics at an early stage. It requires much more mathematical sophistication than most undergraduate students have. (I am talking of US here).
Second, Feynman's integral itself cannot be called fully mathematically justified. There were many attempts to formulate it rigorously, I am not prepared to analyse these attempts and how successful they are, but one thing is clear: this justification requires by far more advanced mathematics than most physicists know (let alone undergraduates). In their daily work most physicists just do not care about rigorous mathematical justification. But teaching is another matter:-)
Third. It is not true that Schrodinger's formulation has only historic value. It is simple and convenient, and there is a rich and rigorous mathematical theory behind it. It is widely used by physicists and mathematicians. (Heisenberg's approach (matrix mechanics) is less popular, but even that has much more to it than "historical significance").
Fourth. Education is EXTREMELY conservative. Let me give another example: multivariate calculus (Stokes theorem and its relatives). The modern exposition is based on the formalism of differential forms of E. Cartan, invented in the beginning of 20-th century. It is much SIMPLER and more powerful than what we teach to undergraduates. Still undergraduate calculus is taught with 19-th century formalism of "vector analysis" almost everywhere (except France). The reason for this is that teachers tend to teach the way they were taught themselves:-)
But in the case of Feynman integral, I think that the main reasons are the second and third.
By the way, Feynman himself, in his Lectures on Physics explains quantum mechanics without his integral. He has one special lecture (a kind of appendix) on the Principle of least action, but warns that this is a much more advanced stuff than the rest of the book:-)
P.S. I do not want to register for the Physics site to answer the question whether "Lagrangian" must be capitalized. But my opinion is this: there are rules of the language. They may be logical or not, we may like them or not. But there are codified rules, and it is better to obey them. In Russian all these words are written with lower case, in German they write riemannsche Flache, but in English: Riemann surface. So in different languages different words in the same expression are capitalized.
P.P.S. For many years I was looking in bookstores the book of Feynman and Hibbs, Quantum mechanics and path integrals. The shelves are always full of "Are you joking Mr. Feynman?" and such, but this book is long out of print... Finally I purchased a Russian translation in a used book store in Ukraine.
For quantum mechanics path integral can be done rigorously using Feynman-Kac formula and analytic continuation, but the approach is by no means elementary. A single look at the Feynman-Kac formula shows why it is not a great place for students to start at. Moreover, understanding where it comes from is itself a chore. While path integral heuristics are intuitively attractive, implementing them in practice - with actual computations - is a different matter.
While path integral is called "integral" even in the context of quantum mechanics it is something more complicated, a pairing between a distribution and a test function. Analytically continued Feynman-Kac formula is impractical for actual calculations, but what it amounts to is solving an initial-boundary problem for the Schrödinger equation. There are many more methods for doing that, separation of variables and Laplace transform for example, and they are more elementary than the theory of measures and distributions on infinite dimensional spaces.
On the other hand, teaching Feynman diagrams early was argued convincingly by John Baez and others, not in the context of QFT, but in the context of computing finite-dimensional integrals perturbatively, see here for an elementary exposition.
